In a filter which is arranged in an exhaust system of an internal combustion engine for trapping and removing particulate matter (PM) in an exhaust gas, it is generally intended to maintain the PM trapping ability of the filter by oxidizing and removing the PM trapped. Thus, when oxidation removal of the trapped PM is carried out in the filter, a rapid temperature gradient occurs in the filter due to the heat generated by the oxidation reaction, thus giving rise to a possibility that cracks and/or melting loss of the filter may be caused. Accordingly, as a technology to suppress the occurrence of cracks, etc., in the filter, there has been developed one described in Patent Document 1, for example. With this technology, in order to make small a temperature gradient in the filter at the time of PM oxidation, in a region in which a mean temperature of the filter is 100 degrees C.-700 degrees C., an oxidative condition of PM in the filter is regulated in such a manner that a rate of rise of the mean temperature in the filter becomes equal to or less than 100 degrees C./minute.
In addition, as shown in Patent Document 2, a sensor for detecting a desired parameter may be arranged in an exhaust system of an internal combustion engine, and a heater for heating the sensor may be arranged in combination therewith for the purpose of such as enhancing the detection accuracy of the sensor. In this case, when moisture has adhered to the sensor at the time the sensor is to be heated by the heater, a crack may occur in the sensor due to a difference in temperature between the sensor side thus heated and the moisture side. Accordingly, in Patent Document 2, there is disclosed a technology in which heating of the sensor by means of the heater is carried out in a gentle or gradual manner, so that a difference in temperature between the interior of the heater and the surface thereof does not exceed a predetermined value.